The exam is due 5.4 8pm. The exam is attached and I will be providing help for any question ragarding the exam

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EE 353

Final Exam

4 May 2020

Last Name (Print):

_______________________________________

First Name (Print):

_______________________________________

PSU User ID (e.g. map244): _______________________________________

Problem Weight

1

12

2

9

3

9

4

9

5

9

6

9

7

9

8

9

9

9

10

16

Bonus

10

Total

100

Score

Instructions:

Complete the solutions in the space provided. If you need additional space, simply include an extra

page in your exam packet with the work; place it immediately following the problem. Circle or otherwise

enclose your final solutions. The clarity of your mathematical analysis is an important part of your work;

unclear analysis or missing intermediate steps can lead to loss of credit even if your solution is correct.

This exam is open book and open notes. You are free to use a calculator.

In the exam, you will occasionally see words in bold, underlined text telling you the way you MUST solve

the problem to receive credit. I expect all your answers to be supported mathematically.

Academic Integrity Statement:

The work on this exam represents my efforts. From the start of the exam to the end of the exam, I did

not communicate with any student or outside party. I understand that exams without a signed

academic integrity statement will receive a grade of zero.

Signature:

_________________________________________________

Problem 1 (12 points)

1. (6 points) A discrete-time signal is created by sampling the continuous-time function

( ) = 5 cos(2 ) − 3 cos �6 + � + 6 sin(14 )

3

every 100 ms. The resulting discrete-time signal is

[ ] = 5 cos[0.2 ] − 3 cos �0.6 + � + 6 sin[1.4 ]

3

If possible, determine the period of both ( ) and [ ]. If it is not possible, explain why based on

your mathematical analysis.

2. (6 points) Given that [ ] = sinc[ ] rect � �, determine if the signal is a power signal, an energy

8

signal, or neither. If it is a power or energy signal, determine the value of the metric.

Problem 2 (9 points)

A periodic function is given by the Fourier Series

∞

1

1

cos(200 )

( ) = + �

100 2

2

=1

The function has a fundamental frequency of 0 = 100 Hz. Assume that the signal is sampled with a

sampling frequency = 2 kHz. Is it possible to recover the signal ( ) from its samples or will aliasing

interfere? Explain your answer in words and mathematically.

Problem 3 (9 points)

Given the system model

1

1

1

[ ] − [ − 1] + [ − 2] = [ ] − [ − 1]

4

4

2

where [−1] = −2 and [−2] = 8. Assume [ ] = [ ]; solve recursively for [ ] for 0 ≤ ≤ 2.

Problem 4 (9 points)

A system has the impulse response

ℎ[ ] = 0.5 cos � � [ ]

3

Using a convolution method, determine the system’s zero state response, [ ], to the input signal

1

6

[ ] = � � [ ].

Problem 5 (9 points)

Given the system model

6 [ ] − 5 [ − 1] = 18 [ ]

2

5

where [0] = 2. Assume [ ] = �− � [ ]. Use the classical solution method to solve the linear

difference equation to find [ ], the particular solution, and [ ], the natural solution, for > 0.

Problem 6 (9 points)

Demonstrate sliding tape convolution to find

[ ] = ℎ[ ] ∗ [ ]

Assume that ℎ[ ] = 7 [ + 1 ] + 4 [ ] and [ ] = 2 [ ] + 9 [ − 2]

Problem 7 (9 points)

Let [ ] = 9 [ ] − 6 [ − 1] + 3 [ − 2 ]. Use the definition of the Discrete Fourier Transform to

find , the spectrum of the signal. In your solution, express complex numbers in polar form using

degrees for your angles.

Problem 8 (8 points)

A system has the impulse response

ℎ[ ] = �0.2 + (−0.3) � [ ]

Use the summation definition of the Z-transform to find the transfer function of the system. Express

your solutions as a ratio of polynomials in z with the denominator in standard form.

Problem 9 (9 points)

Given the system model

[ ] − 0.3 [ − 1] + 0.02 [ − 2] = 2 [ ] − 3 [ − 1]

2

3

where [−1] = 2 and [−2] = 1. Assume [ ] = � � [ ]. Solve the difference equation using the

delay property of the Z-transform to determine the zero state solution [ ] and the zero input

solution [ ]. Express your solutions as a ratio of polynomials in z with the denominator in standard

form. Do not solve for [ ].

Problem 10 (16 points)

The Z-transform of the output of a system is given by

6 4 − 6.6 3 + 1.67 2 + 0.105

[ ] =

( + 0.3)( − 0.6)( − 0.5)2

Use the method of partial fraction expansion to determine the inverse Z-transform of [ ].

Bonus Problem (10 points)

Figure 1 shows a system block diagram.

Figure 1. The system block diagram for the bonus problem.

Assume the proportional gain is = 2, the plant transfer function is [ ] =

transfer function is [ ] =

.

+0.3

and

+2

the sensor

Find the overall transfer function of the system, [ ], so that

[ ] = [ ] [ ]

Express your solutions as a ratio of polynomials in z with the denominator in standard form. Determine

mathematically if the resulting system is asymptotically stable.

…

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